A method of solving real linear homogeneous second-order differential equations in an infinite interval (Q1803491)

The author considers an approximate method for the solution of the differential equation \(y''+p(x)y=0\) on the interval \((0,\infty)\). Suppose that the coefficient \(p(x)\) has the limiting value \(p_ 0\) at \(\infty\), and its asymptotic expansion is known as \(x\to\infty\). She gives a systematic way to derive the asymptotic expansion of the solution. The initial idea is to consider the new unknown \(A(x)=Y'(x)/Y(x)\) where, putting \(y_ 1(x)\) and \(y_ 2(x)\) as bounded solutions, \(Y(x)=y_ 1(x)+iy_ 2(x)\). The introduction of the complex-valued function makes the problem harder, but it is claimed the method is more useful. A simple example is given.